control of hybrid systemsmaler/tcc/morari-tcc.pdfconstrained finite time optimal control of pwa...

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Automatic Control Laboratory, ETH Zürichwww.control.ethz.ch

Control of Hybrid SystemsTheory, Computation & Applications

Manfred Morari

M. Baotic, F. Christophersen,T. Geyer, P. Grieder, C. Jones. M. Kvasnica,

U. Mäder, D. Niederberger, G. Papafotiou

Outline

1. Basics• Hybrid Modeling• Controller Computation

2. Low Complexity Controllers• The Three Levers of Complexity• Minimum-Time Controller• N-Step Controller

3. Industrial Applications• Control of a DC-DC Converter• Vibration Damping

4. Conclusions

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Outline




4. Conclusions

•Piecewise affine (PWA) systems.•Polyhedral partition of the state space.•Affine dynamics in each region.

PWA Hybrid Models

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Equivalence of Hybrid Models Equivalent classes of discrete-time hybrid models:

• Mixed Logical Dynamical (MLD)• Piecewise affine (PWA)• Linear Complementarity (LC)• Extended Linear Complementarity (ELC)• Max-Min-Plus-Scaling (MMPS)

The above discrete-time hybrid models can be proven to beequivalent (under some mild assumptions). Equivalenceimplies that for all feasible initial states and for all feasibleinput trajectories, the models yield the same state and outputtrajectories.

[Heemels, De Schutter, Bemporad, 2001]

Outline




4. Conclusions

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System• Discrete PWA Dynamics• Constraints on the state x(k) 2 X• Constraints on the input u(k) 2 U

Objectives• Stability (feedback is stabilizing)• Feasibility (feedback exists for all time)• Optimal Performance

Optimal Control forConstrained PWA Systems

Constrained Finite Time OptimalControl of PWA Systems

Algebraicmanipulation

Mixed Integer Linear Program (MILP)

Linear Performance Index (p=1,1)

Constraints

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Constrained Finite Time OptimalControl of PWA Systems

Algebraicmanipulation


Constraints

Receding Horizon Control

Mixed Integer Linear Program (MILP)

PLANT output y

plant state x

apply u0*

OptimizationProblem (MILP)obtain U*(x)

Receding Horizon ControlOn-Line Optimization

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OptimizationProblem (mpMILP)

Explicit Solution

u* = f(x)

PLANT output y

plant state xcontrol u*

(=Look-Up Table)

Receding Horizon Policy: Off-Line Opt.Multi-Parametric Programming

off-line

Characteristics of mp-MILP Solution

k Qxk{1, 1 } CR is polyhedral

• The optimizer z*(x) is piecewise affine• The feasible set X* is partitioned into critical regions

(CR).• The value function J*(x) is piecewise affine.

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Receding Horizon ControlStability Guarantee


Stability Guarantee

• Choose horizon T= ∞=> Constrained Infinite Time Optimal Control

or

• Choose ||PxT||p to be a (Control) Lyapunov Function

Receding Horizon ControlStability Guarantee


Stability GuaranteeChoose ||PxT||p to be a (Control) Lyapunov Function => Simple LP or algebraic test Christophersen, 2006

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Multi-parametric controllersAlgorithms have been developed for over 5 years:

…Minimization of linear and quadratic objectives (Baotic, Baric, Bemporad, Borrelli, De Dona, Dua, Goodwin, Grieder,

Johansen, Mayne, Morari, Pistikopoulos, Rakovic, Seron, Toendel)

…Minimum-Time controller computation(Baotic, Grieder, Kvasnica, Mayne, Morari, Schroeder)

…Infinite horizon controller computation(Baotic, Borrelli, Christophersen, Grieder, Morari, Torrisi)

…Computation of robust controllers(Borrelli, Bemporad, Kerrigan, Grieder, Maciejowski, Mayne, Morari, Parrilo, Sakizlis)

) Advanced computation schemes are available !

PROs:– Easy to implement– Fast on-line evaluation– Analysis of closed-loop system possible

CONs:– Number of controller regions can be large– Off-line computation time may be prohibitive– Computation scales badly.

) controller complexity is the crucial issue

Multi-parametric controllers

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Outline




4. Conclusions

ControllerConstruction

Partition Complexity

PLANT Output y

State xControl u*

3 Complexity Levers ofReceding Horizon Control

off-line

on-line

1

3

2

Region Identification

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ObjectiveCompute controller partition as quickly as possible.

Why is computation time an issue?– For LTI systems, controller computation time correlates to

controller complexity– For PWA systems, controller computation time may not

correlate to controller complexity

However…– Controller complexity and runtime grows exponentially with

problem size

) Controller computation is not a bottleneck

1st Lever forComplexity Reduction

Optimal merging:hyperplane arrangmt: 34 secmerging (boolean min.): 5 sec

2nd Lever forComplexity Reduction - Region Merging

189 polyhedra

252 polyhedra

39 polyhedra

Greedy merging:merging based on LPs: 17 sec

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ObjectiveFor a given partition, identify controller region as quickly aspossible.

AlgorithmsIdentification of region in time logarithmic in the number ofregions is possible (Bemporad, Grieder, Johansen, Jones, Rakovic,Toendel)

However…– Pre-processing for most schemes is very expensive– Usually not applicable to general partitions (e.g. overlapping

regions, holes, non-convex partitions, etc.)

3rd Lever forComplexity Reduction

IdeaConstruct an interval tree based onbounding boxes. The search thenreduces to answering stabbingqueries in O(log(N)) time.

Advantage– Very cheap pre-processing.– Applicable to any type of partitions (not nec. polyhedral).

On-line strategy– The tree selects possible candidate regions.– Locally search the list of candidates.

Region Identification usingBounding Boxes

(Christophersen et. al, submitted to CDC 2006)

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Control Objective vs.Partition Complexity

Objective– Compute controller partition with as few regions as possible– Guarantee stability and constraint satisfaction

Observation Complex objectives yield complex controllers

Approach Use simpler objectives to obtain simpler controllers

) Controller complexity is a bottleneck

Outline




4. Conclusions

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Numerical ExampleN-Step ControlOptimal Control

• All results and plots were obtained with the MPTtoolbox

http://control.ethz.ch/~mpt

• MPT is a MATLAB toolbox that provides efficientcode for– (Non)-Convex Polytope Manipulation– Multi-Parametric Programming– Control of PWA and LTI systems

Conclusions

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Outline




4. Conclusions

Furuta Pendulum• Rotating inverted

pendulum• Explicit controller

with 5 regions• Results equivalent to [Åkesson and Åström, 2001]

[Akesson and Aström, 2001]

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Switch-mode DC-DC Converter

d

unregulated DC voltage low-pass filter load

dually operated switches

Switched circuit: supplies power to load with constant DC voltage

Illustrating example: synchronous step-down DC-DC converter

regulated DC voltage

S1

S2

Operation Principle

k-th switching period

duty cycle

• Length of switching period Ts constant (fixed switching frequency)• Switch-on transition at kTs, k2N • Switch-off transition at (k+d(k))Ts (variable pulse width)• Duty cycle d(k) is real variable bounded by 0 and 1

S1 = 1S2 = 0

S1 = 0S2 = 1

kTs (k+1)Ts(k+d(k))Ts

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S1 = 1S2 = 0

S1 = 0S2 = 1

kTs (k+1)Ts(k+d(k))Ts

duty cycle d(k)

Mode 1 and 2

mode 1: mode 2:

d

S1

S2

Control Objective

unregulated DC input voltagedisturbance

duty cyclemanipulated variable

regulated DC output voltagecontrolled variable

inductor currentstate

capacitor voltagestate

Regulate DC output voltage by appropriate choice of duty cycle

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State-feedback Controller:Polyhedral Partition

Colors correspond to the 121 polyhedra

PWA state-feedbackcontrol law:

computed in 100s using theMPT toolbox

121 polyhedra after simplificationwith optimal merging

Outline




4. Conclusions

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Smart Damping Materials

– Device suppresses vibration– External power source for operation is not required– Weight and size of the device have to be kept to a

minimum

• Demands

• Idea– Switched Piezoelectric (PZT)

Patches• Problem– What is the optimal switching law for optimal vibration

suppression?

PZTPZT

Niederberger, Moheimani

Application:Brake Squeal Reduction

• Friction induced vibration in brakes– Strong vibration radiates unwanted noise– One frequency, small bandwidth– Frequency can vary

Neubauer, Popp

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Brake Squeal Reductionusing Shunt Control

• Vibration reduction– Piezoelectric actuator between

brake pad and calliper– Switching shunt control

• Advantages– Tracks resonance frequency– Cheap solution– No electrical power required

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• Control / Scheduling of Cogeneration Power Plants

• Control / Scheduling of Cement Kilns and Mills

• Electronic Throttle Control / Traction Control

• Control of Thermal Print Heads

• Control of Voltage Stability for Electric PowerNetworks

• Direct Torque Control of 3 phase Induction Motors

• Control of Anaesthesia (Inselspital Bern)

• Adaptive Cruise Control

Other Applications ofHybrid Systems Control Tools

• All results and plots were obtained with the MPTtoolbox

http://control.ethz.ch/~mpt

• MPT is a MATLAB toolbox that provides efficientcode for– (Non)-Convex Polytope Manipulation– Multi-Parametric Programming– Control of PWA and LTI systems

Conclusions

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Facts about MPT

4000+ downloads

Rated 4.5 / 5 on mathworks.com

Conclusions

• Foundations of a theoretical framework forfor practical controller design for PWAsystem have been established

• Complexity reduction and robustness aremain research issues

• Applications in industry are beginning

control of hybrid systemsmaler/tcc/morari-tcc.pdfconstrained finite time optimal control of pwa...

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